论文标题
编织的表面及其特征图
Braided surfaces and their characteristic maps
论文作者
论文摘要
我们表明，足够大属的表面的分支覆盖物是编织表面的特征图，即带有$ \ mathbb r^2 $的表面嵌入。这个结果对于未受到覆盖的覆盖物来说已经是不平凡的，在这种情况下，提起问题是众所周知的，可以减少将单莫莫·图（Monodromy Map）分解为对称组$ s_n $的纯粹代数问题。在我们的方法中，这种分解通常是由于更强的特性而实现的：通过自由群体进行分解。在相反的方向上，我们表明，任何非亚洲表面群体都具有许多有限的简单的非亚伯利亚群体，具有特征性内核，这些核不包含任何简单的循环，因此商图不会通过自由组来考虑。通过撤回构造，有限的尺寸赫米尔族人代表辫子为编织的表面提供不变的。我们表明，当且仅当它们被准确地分开时，编织表面的强质量类别被这种不变性分开。
We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for unramified coverings, in which case the lifting problem is well-known to reduce to the purely algebraic problem of factoring the monodromy map to the symmetric group $S_n$ through the braid group $B_n$. In our approach, this factorization is often achieved as a consequence of a stronger property: a factorization through a free group. In the reverse direction we show that any non-abelian surface group has infinitely many finite simple non-abelian groups quotients with characteristic kernels which do not contain any simple loop and hence the quotient maps do not factor through free groups. By a pullback construction, finite dimensional Hermitian representations of braid groups provide invariants for the braided surfaces. We show that the strong equivalence classes of braided surfaces are separated by such invariants if and only if they are profinitely separated.
